Consider the monthly adjusted closing prices of Apple Inc. (AAPL), Amazon.com Inc. (AMZN), Wal-Mart Stores Inc. (WMT) and Exxon Mobil Corp. (XOM) from February 1, 2012 to December 29, 2017.

	Price					Return
Date	AAPL	AMZN	WMT	XOM	S&P 500	AAPL	AMZN	WMT	XOM	S&P 500
2/1/2012	52.26	179.69	49.986584	70.4	1365.68
3/1/2012	57.76	202.51	51.780273	70.97	1408.47	10.52%	12.70%	3.59%	0.81%	3.13%
4/1/2012	56.26	231.9	50.175938	70.65	1397.91	-2.60%	14.51%	-3.10%	-0.45%	-0.75%
5/1/2012	55.66	212.91	56.061443	64.34	1310.33	-1.07%	-8.19%	11.73%	-8.93%	-6.27%
6/1/2012	56.26	228.35	59.785698	70.5	1362.16	1.08%	7.25%	6.64%	9.57%	3.96%
7/1/2012	58.84	233.3	63.824551	71.56	1379.32	4.59%	2.17%	6.76%	1.50%	1.26%
8/1/2012	64.09	248.27	62.25531	71.93	1406.58	8.92%	6.42%	-2.46%	0.52%	1.98%
9/1/2012	66.25	254.32	63.626152	75.84	1440.67	3.37%	2.44%	2.20%	5.44%	2.42%
10/1/2012	59.12	232.89	64.677956	75.61	1412.16	-10.76%	-8.43%	1.65%	-0.30%	-1.98%
11/1/2012	58.12	252.05	62.091534	73.09	1416.18	-1.69%	8.23%	-4.00%	-3.33%	0.28%
12/1/2012	54.59	250.87	58.824013	72.22	1426.19	-6.07%	-0.47%	-5.26%	-1.19%	0.71%
1/1/2013	46.72	265.5	60.641113	75.08	1498.11	-14.42%	5.83%	3.09%	3.96%	5.04%
2/1/2013	45.28	264.27	61.360676	74.73	1514.68	-3.08%	-0.46%	1.19%	-0.47%	1.11%
3/1/2013	47.32	266.49	64.871719	75.68	1569.19	4.51%	0.84%	5.72%	1.27%	3.60%
4/1/2013	47.34	253.81	67.811798	74.73	1597.57	0.04%	-4.76%	4.53%	-1.26%	1.81%
5/1/2013	48.08	269.2	65.298958	75.98	1630.74	1.56%	6.06%	-3.71%	1.67%	2.08%
6/1/2013	44.44	277.69	65.383408	76.4	1606.28	-7.57%	3.15%	0.13%	0.55%	-1.50%
7/1/2013	50.71	301.22	68.411636	79.28	1685.73	14.11%	8.47%	4.63%	3.77%	4.95%
8/1/2013	54.6	280.98	64.058014	73.7	1632.97	7.67%	-6.72%	-6.36%	-7.04%	-3.13%
9/1/2013	56	312.64	65.312408	73.26	1681.55	2.56%	11.27%	1.96%	-0.60%	2.97%
10/1/2013	61.4	364.03	67.776199	76.31	1756.54	9.64%	16.44%	3.77%	4.16%	4.46%
11/1/2013	65.32	393.62	71.538116	79.59	1805.81	6.38%	8.13%	5.55%	4.30%	2.80%
12/1/2013	68.69	398.79	69.48938	86.75	1848.36	5.16%	1.31%	-2.86%	9.00%	2.36%
1/1/2014	61.29	358.69	66.33213	79	1782.59	-10.77%	-10.06%	-4.54%	-8.93%	-3.56%
2/1/2014	64.43	362.1	66.349876	82.53	1859.45	5.12%	0.95%	0.03%	4.47%	4.31%
3/1/2014	68.57	336.37	67.886513	84.33	1872.34	6.43%	-7.11%	2.32%	2.18%	0.69%
4/1/2014	75.39	304.13	71.25663	88.41	1883.95	9.95%	-9.58%	4.96%	4.84%	0.62%
5/1/2014	80.87	312.55	68.628426	86.79	1923.57	7.27%	2.77%	-3.69%	-1.83%	2.10%
6/1/2014	86.47	324.78	67.524185	87.51	1960.23	6.92%	3.91%	-1.61%	0.83%	1.91%
7/1/2014	88.95	312.99	66.183952	86	1930.67	2.87%	-3.63%	-1.98%	-1.73%	-1.51%
8/1/2014	95.37	339.04	67.910957	86.45	2003.37	7.22%	8.32%	2.61%	0.52%	3.77%
9/1/2014	94.21	322.44	69.236626	82.32	1972.29	-1.22%	-4.90%	1.95%	-4.78%	-1.55%
10/1/2014	100.99	305.46	69.055511	84.64	2018.05	7.20%	-5.27%	-0.26%	2.82%	2.32%
11/1/2014	111.21	338.64	79.259499	79.24	2067.56	10.12%	10.86%	14.78%	-6.38%	2.45%
12/1/2014	103.66	310.35	77.756493	81.5	2058.9	-6.79%	-8.35%	-1.90%	2.85%	-0.42%
1/1/2015	110.03	354.53	77.371483	77.07	1994.99	6.15%	14.24%	-0.50%	-5.44%	-3.10%
2/1/2015	120.64	380.16	76.415489	78.05	2104.5	9.64%	7.23%	-1.24%	1.27%	5.49%
3/1/2015	117.32	372.1	74.885902	75.5	2067.89	-2.75%	-2.12%	-2.00%	-3.27%	-1.74%
4/1/2015	118	421.78	71.48877	77.6	2085.51	0.58%	13.35%	-4.54%	2.78%	0.85%
5/1/2015	122.84	429.23	68.026535	75.67	2107.39	4.10%	1.77%	-4.84%	-2.49%	1.05%
6/1/2015	118.76	434.09	65.377335	74.51	2063.11	-3.32%	1.13%	-3.89%	-1.53%	-2.10%
7/1/2015	114.85	536.15	66.345139	70.94	2103.84	-3.29%	23.51%	1.48%	-4.79%	1.97%
8/1/2015	106.91	512.89	59.662697	67.38	1972.18	-6.91%	-4.34%	-10.07%	-5.02%	-6.26%
9/1/2015	104.91	511.89	60.172169	67.21	1920.03	-1.87%	-0.19%	0.85%	-0.25%	-2.64%
10/1/2015	113.66	625.9	53.119293	74.79	2079.36	8.34%	22.27%	-11.72%	11.28%	8.30%
11/1/2015	112.51	664.8	54.604107	73.81	2080.41	-1.01%	6.22%	2.80%	-1.31%	0.05%
12/1/2015	100.54	675.89	56.887009	71.08	2043.94	-10.64%	1.67%	4.18%	-3.70%	-1.75%
1/1/2016	92.98	587	62.098557	70.98	1940.24	-7.52%	-13.15%	9.16%	-0.14%	-5.07%
2/1/2016	92.35	552.52	62.079849	73.08	1932.23	-0.68%	-5.87%	-0.03%	2.96%	-0.41%
3/1/2016	104.67	593.64	64.091789	76.91	2059.74	13.34%	7.44%	3.24%	5.24%	6.60%
4/1/2016	90.02	659.59	63.03907	81.34	2065.3	-14.00%	11.11%	-1.64%	5.76%	0.27%
5/1/2016	95.9	722.79	66.725052	81.91	2096.95	6.53%	9.58%	5.85%	0.70%	1.53%
6/1/2016	92.37	715.62	69.340729	86.97	2098.86	-3.68%	-0.99%	3.92%	6.18%	0.09%
7/1/2016	100.69	758.81	69.293259	82.53	2173.6	9.01%	6.04%	-0.07%	-5.11%	3.56%
8/1/2016	102.51	769.16	67.84034	80.85	2170.95	1.81%	1.36%	-2.10%	-2.04%	-0.12%
9/1/2016	109.82	837.31	68.954903	81.67	2168.27	7.13%	8.86%	1.64%	1.01%	-0.12%
10/1/2016	110.3	789.82	66.947067	77.96	2126.15	0.44%	-5.67%	-2.91%	-4.54%	-1.94%
11/1/2016	107.36	750.57	67.339088	81.69	2198.81	-2.67%	-4.97%	0.59%	4.78%	3.42%
12/1/2016	113.09	768.66	66.086578	85.21	2238.83	5.34%	2.41%	-1.86%	4.31%	1.82%
1/1/2017	118.49	823.48	64.267723	79.19	2278.87	4.77%	7.13%	-2.75%	-7.06%	1.79%
2/1/2017	133.76	845.04	68.302513	76.77	2363.64	12.89%	2.62%	6.28%	-3.06%	3.72%
3/1/2017	140.88	886.54	69.409912	78.13	2362.72	5.32%	4.91%	1.62%	1.77%	-0.04%
4/1/2017	140.87	924.99	72.927391	77.78	2384.2	-0.01%	4.34%	5.07%	-0.45%	0.91%
5/1/2017	149.8	994.62	76.244934	76.69	2411.8	6.34%	7.53%	4.55%	-1.40%	1.16%
6/1/2017	141.82	968	73.903694	77.63	2423.41	-5.33%	-2.68%	-3.07%	1.23%	0.48%
7/1/2017	146.45	987.78	78.112534	76.97	2470.3	3.26%	2.04%	5.70%	-0.85%	1.93%
8/1/2017	161.49	980.6	76.237595	73.4	2471.65	10.27%	-0.73%	-2.40%	-4.64%	0.05%
9/1/2017	152.36	961.35	76.785919	79.6	2519.36	-5.65%	-1.96%	0.72%	8.45%	1.93%
10/1/2017	167.11	1105.28	85.797005	80.93	2575.26	9.68%	14.97%	11.74%	1.67%	2.22%
11/1/2017	169.88	1176.75	95.545113	80.87	2584.84	1.66%	6.47%	11.36%	-0.07%	0.37%
12/1/2017	167.9	1169.47	97.038773	81.96	2673.61	-1.17%	-0.62%	1.56%	1.35%	3.43%
Mean	93.9487	511.99493	67.397028	77.4923	1942.2379	1.90%	2.99%	1.07%	0.31%	1.00%
Stand. Dev	34.7401	277.21649	8.4573782	5.31729	348.67321	6.71%	7.60%	4.87%	4.24%	2.76%
Covariance Matrix						aapl	amz	wmt	xon	sp
	AAPL	AMZN	WMT	XOM	S&P 500
AAPL	0.0045	0.0016841	0.0005633	0.00022	0.0009108
AMZN	0.00168	0.005772	0.0001184	0.00058	0.0011002
WMT	0.00056	0.0001184	0.0023758	1.5E-05	0.0001161
XOM	0.00022	0.0005785	1.478E-05	0.0018	0.0006736
S&P 500	0.00091	0.0011002	0.0001161	0.00067	0.0007616

1. Suppose that at the end of the trading day on January 3, 2018, you want to build a portfolio with these four stocks using the monthly statistical parameters from above. Let r = (rAAPL + rAMZN + rWMT + rXOM) / 4 be the average of the expected monthly rates of return of these stocks. Find the weights of the minimum variance portfolio with expected rate of return r. Moreover, if on that date you have $1,000,000, deduce the number of shares of each of these stocks you will have in your portfolio.

2. Using the adjusted closing prices on June 29, 2018 find the market value of your portfolio and the rate of return of your investment.

3. Using the S&P 500 as the Market and the same historical data (monthly adjusted closing prices from February 1, 2012 to December 29, 2017). Compute the Beta of each of these stocks.

4. If the monthly risk-free rate is rf = 0.0017, compute the Jensen and Sharpe indexes of each of these stocks. Derive the Jensen and Sharpe indexes of your portfolio. What can you conclude?